
Conditionally on \(\left\lbrace X_2=j\right\rbrace \) , \(X_1\)  and \(X_3\)  are distributed
independently, each as the sum of \(j\)  independent random variables with
generating function \(p(s)\) , plus one with generating function
\(\varphi (s,1\mid 0)\) — so \(X_t\)  is a branching process [1]}
whose unconditional three-dimensional marginal distributions have generating
function:
\(\varphi ({s},{z},{t}) &:= {\textsf {E}}\big [{s}^{X_1}{z}^{X_2}{t}^{X_3}\big ]\\&= {\varphi ({s},{t}\mid 0)} \sum _{j\ge 0} {z}^j p({s})^j p({t})^j {\textsf {P}}[X_2=j]\\&= {\varphi ({s},{t}\mid 0)} {\textsf {E}}\left[ z p({s}) p({t})\right]^{X_2}\\&= {\varphi ({s},{t}\mid 0)} \varphi \big ( z p({s}) p({t})\big )\\&= {\varphi ({s},{t}\mid 0)} \exp \big [P\big ( z p({s}) p({t})\big )-{\theta _{+}}\big ].\) 
